In recent years, the integration of robotic arms with quadrotor unmanned aerial vehicles (UAVs) has garnered significant attention due to its potential in applications such as aerial manipulation, inspection, and payload delivery. However, the dynamic coupling between the quadrotor and the robotic arm introduces challenges in stability and control, especially under external disturbances and model uncertainties. This paper addresses these issues by developing a comprehensive dynamics model and a dual-controller architecture. We employ the Euler-Lagrange formulation to derive the system dynamics and design a linear active disturbance rejection controller (LADRC) for position and attitude control, combined with an adaptive controller based on RBF neural network compensation for joint control. The proposed approach minimizes reliance on precise model knowledge and enhances robustness. Extensive simulations validate the effectiveness of our method in various scenarios, demonstrating superior performance compared to conventional PID control.
The quadrotor system with a robotic arm is a complex multi-body system. We consider a quadrotor equipped with a two-joint robotic arm, as illustrated in the figure below. The system consists of the quadrotor base and the robotic arm, which introduces additional degrees of freedom and dynamic interactions. The inertial frame \( O_e \) and the body frame \( O_b \) are defined to describe the system’s motion. The robotic arm has joint angles denoted by \( Q = [q_1, q_2]^T \), and the quadrotor’s position in the inertial frame is \( p = [x, y, z]^T \). The orientation is represented by Euler angles \( \Phi = [\phi, \theta, \psi]^T \), corresponding to roll, pitch, and yaw, respectively. The generalized coordinates for the entire system are given by:
$$ H = [p^T, \Phi^T, Q^T]^T $$
The rotation matrix from the body frame to the inertial frame, \( R \), is expressed as:
$$ R = \begin{bmatrix}
\cos\theta\cos\psi & \cos\psi\sin\theta\sin\phi – \sin\psi\cos\phi & \cos\psi\sin\theta\cos\phi + \sin\psi\sin\phi \\
\cos\theta\sin\psi & \sin\psi\sin\theta\sin\phi + \cos\psi\cos\phi & \sin\psi\sin\theta\cos\phi – \cos\psi\sin\phi \\
-\sin\theta & \sin\phi\cos\theta & \cos\phi\cos\theta
\end{bmatrix} $$
We use the Euler-Lagrange equations to derive the dynamics. The Lagrangian \( L \) is defined as the difference between kinetic energy \( K \) and potential energy \( U \):
$$ L = K – U $$
The equations of motion are:
$$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) – \frac{\partial L}{\partial q} = \tau + \tau_e $$
where \( \tau \) is the generalized force from system inputs, and \( \tau_e \) represents external disturbances. The total kinetic energy \( K \) includes contributions from the quadrotor base and the robotic arm links:
$$ K = K_b + \sum_{i=1}^{2} K_{ci} $$
Here, \( K_b \) is the kinetic energy of the quadrotor, and \( K_{ci} \) is the kinetic energy of the i-th link. The potential energy \( U \) accounts for gravitational effects:
$$ U = m_b g e^T p + \sum_{i=1}^{2} m_{ci} g e^T (p + R p_{bci}) $$
where \( m_b \) and \( m_{ci} \) are masses, \( g \) is gravity, and \( e = [0, 0, 1]^T \). The resulting dynamics form a second-order nonlinear system:
$$ M(q) \ddot{q} + C(q, \dot{q}) \dot{q} + G(q) = \tau + \tau_e $$
where \( M(q) \) is the mass matrix, \( C(q, \dot{q}) \) is the Coriolis matrix, and \( G(q) \) is the gravity vector. This model captures the coupling between the quadrotor and the robotic arm.
To address the control challenges, we propose a dual-controller architecture. For the quadrotor’s position and attitude control, we employ LADRC, which is robust to disturbances and model uncertainties. For the robotic arm joints, we use an adaptive controller with RBF neural network compensation. The LADRC design for a second-order system considers the plant:
$$ \ddot{y} = a \dot{y} + c y + b u + \zeta $$
where \( u \) is the input, \( y \) is the output, and \( \zeta \) is disturbance. By lumping all uncertain terms into a total disturbance \( f(y, \dot{y}, \zeta) \), we rewrite the system as:
$$ \ddot{y} = b_0 u + f(y, \dot{y}, \zeta) $$
A linear extended state observer (LESO) estimates the states and disturbance:
$$ \begin{aligned}
e &= z_1 – y \\
\dot{z}_1 &= z_2 – \beta_{01} e \\
\dot{z}_2 &= z_3 – \beta_{02} e + b_0 u \\
\dot{z}_3 &= -\beta_{03} e
\end{aligned} $$
where \( z_1, z_2, z_3 \) estimate \( y, \dot{y}, f \), respectively. The observer gains are set as \( [\beta_{01}, \beta_{02}, \beta_{03}]^T = [3\omega_0, 3\omega_0^2, \omega_0^3]^T \). The control law is:
$$ u = \frac{u_0 – z_3}{b_0} $$
with a PD controller:
$$ u_0 = -k_p (z_1 – r) – k_d z_2 $$
where \( r \) is the reference, and \( [k_p, k_d]^T = [\omega_c^2, 2\omega_c]^T \). This structure enables disturbance rejection without precise model knowledge.
For the robotic arm, we design an adaptive controller based on RBF neural network compensation. The nominal dynamics are:
$$ M_e(q) \ddot{q} + C_e(q, \dot{q}) \dot{q} + G_e(q) = \tau + f(q, \dot{q}, \ddot{q}) $$
where \( f \) represents model errors and disturbances. The RBF network approximates \( f \) as:
$$ \hat{f} = w^T h(x) $$
with Gaussian basis functions \( h_j = \exp\left( \frac{\|x – c_j\|^2}{b_j^2} \right) \). The control input for the arm is:
$$ \tau = M_e(q) (\ddot{q}_d – k_1 \dot{e} – k_2 e) + C_e(q, \dot{q}) \dot{q} + G_e(q) – \hat{f} $$
where \( e = q – q_d \) is the tracking error. The adaptation law for the network weights is:
$$ \dot{\hat{w}}^T = \gamma B^T P x h^T $$
ensuring stability via Lyapunov analysis. This approach compensates for uncertainties in real-time.
We conducted simulations in MATLAB/Simulink to evaluate the proposed control strategy. The system parameters are listed in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \( m_b \) (kg) | 1.40 | \( b \) (m) | 0.03 |
| \( m_{m1} \) (kg) | 0.16 | \( c_t \) (Ns²) | 7.1e-5 |
| \( m_{m2} \) (kg) | 0.046 | \( c_m \) (Nms²) | 1.6e-6 |
| \( L_1 \) (m) | 0.12 | \( I_{xx} \) (kg·m²) | 1.31e-2 |
| \( L_2 \) (m) | 0.056 | \( I_{yy} \) (kg·m²) | 1.31e-2 |
| \( c \) (m) | 0.066 | \( I_{zz} \) (kg·m²) | 1.47e-2 |
The controller parameters for LADRC and RBF adaptive control are summarized in Table 2.
| Channel | Symbol | Parameters |
|---|---|---|
| Roll | \( \omega_0, \omega_1, b_0 \) | 25, 15, 0.35 |
| Pitch | \( \omega_0, \omega_1, b_0 \) | 25, 15, 0.35 |
| Yaw | \( \omega_0, \omega_1, b_0 \) | 20, 10, 0.10 |
| X | \( \omega_0, \omega_1, b_0 \) | 3, 0.9, 15.00 |
| Y | \( \omega_0, \omega_1, b_0 \) | 3, 0.9, 15.00 |
| Z | \( \omega_0, \omega_1, b_0 \) | 13, 2.3, 0.10 |
| Joint Angle | \( k, \gamma \) | 1.5, 20.00 |
| RBF Centers | \( c \) | [-3, -2, -1, 0, 1, 2, 3] |
In the first experiment, the robotic arm is static with fixed joint angles \( q_1 = 0.5 \) rad and \( q_2 = 0.5 \) rad, and the quadrotor is commanded to reach a desired position \( p_b = [2, 2, 3]^T \) m. The results show that our controller achieves faster convergence and lower steady-state error compared to PID. For instance, the position errors in x, y, and z axes converge to zero within 8 s, 10 s, and 4 s, respectively, whereas PID exhibits overshoot and longer settling times (11 s, 15 s, 7 s).
The second experiment involves dynamic motion of the robotic arm, with desired joint angles \( q_{1d} = 0.75 + 0.75 \sin t \) rad and \( q_{2d} = 0.75 – 0.75 \sin t \) rad, while the quadrotor maintains position \( p_b = [2, 2, 3]^T \) m. Our method demonstrates superior tracking performance for both the quadrotor position and arm joints, with minimal oscillations and errors. In contrast, PID control shows significant overshoot and slower response, highlighting the effectiveness of our approach in handling dynamic coupling.
To test disturbance rejection, we introduced a step disturbance torque of 3 N·m and model uncertainties of 20% in the dynamics. The RBF-based adaptive controller effectively compensates for these perturbations, maintaining accurate joint tracking. PID control, however, exhibits noticeable deviations and slower recovery. This underscores the robustness of our neural network compensation in adverse conditions.
The stability analysis for the adaptive controller is based on Lyapunov theory. Consider the Lyapunov function:
$$ V = \frac{1}{2} x^T P x + \frac{1}{2\gamma} \|\tilde{w}\|^2 $$
where \( \tilde{w} \) is the weight error. Its derivative satisfies:
$$ \dot{V} \leq -\frac{1}{2} \lambda_{\min}(Q) \|x\|^2 + \|\eta^T\| \|B\| \lambda_{\max}(P) \|x\| $$
ensuring convergence when \( \|x\| \geq \frac{2 \|B\| \lambda_{\max}(P) \|\eta^T\|}{\lambda_{\min}(Q)} \). This guarantees bounded tracking errors under uncertainties.
In summary, we have developed a comprehensive framework for modeling and controlling a quadrotor UAV with a robotic arm. The dynamics model derived via Euler-Lagrange equations accurately captures the system’s behavior. The dual-controller architecture, combining LADRC for the quadrotor and RBF-based adaptive control for the arm, achieves high performance with minimal model dependency. Simulations under static, dynamic, and disturbed conditions validate the robustness and efficiency of our approach. Future work will focus on experimental validation and extension to more complex manipulators.
The integration of robotic arms with quadrotor systems opens new avenues for aerial robotics. Our control strategy effectively addresses the challenges of coupling and disturbances, making it suitable for real-world applications. The use of LADRC and neural networks provides a scalable solution for various quadrotor-based systems, ensuring reliability in unpredictable environments.